In practice, its very difficult to receive a complex signal and accurately demodulate the complex signal into its inphase and quadrature components. Consider for example a radio frequency (RF) sinusoidal signal, e.g., an RF carrier. In theory, a complex receiver should be able to represent such a sinusoidal signal by its inphase, real component and its quadrature, imaginary component. The magnitude of the inphase and quadrature components should be the same, and they should always be separated by a constant 90.degree. phase difference.
The sinusoidal signal may be defined mathematically by v=V cos (.omega.t+.phi.), where if the frequency .omega. is known, then v is specified by its magnitude V and by its phase .phi. in the complex domain, i.e., v=Ve.sup.j.phi.. The complex term, e.sup.j.phi., may be expanded as e.sup.j.phi. =cos .phi.+j sin .phi. using Euler's equation. Euler's equation reveals the ideal balance between inphase and quadrature components of a complex signal. The magnitude of both the cosine and sine terms is exactly the same, i.e., "1", and the cosine and sine terms are exactly 90.degree. out of phase.
The difficulty is trying to achieve the theoretical--Euler's formula--in an imperfect real world. In practice, the phase difference between the inphase and quadrature components is something other than 90.degree., and the magnitudes of the real and imaginary components of the complex signal are not the same. As a result, a demodulated complex signal is no longer accurately represented by its inphase and quadrature components. For ease of reference, the amount by which the phase difference between the inphase and quadrature components is something other than 90.degree. and/or the amount by which the magnitudes of the real and imaginary components of the complex signal are not the same are (is) referred to either in combination or alone as "quadrature imbalance". To date, there is has not been any satisfactory practical implementation in a radio transceiver which would detect and compensate for such undesirable quadrature imbalances.